What you're describing is a symptom of the gyro not being up to speed, though you will always get some effect of the centrifugal force acting on the pendulum weight. (theres a favourite CPL exam question about getting a climbing turn to the right indication when accelerating down the runway).
Most common reasons in my experience, are low vacuum - especially if you're venturi powered, blocked filter and lastly worn out gyro bearings.
As said before the filter is the quickest and easiest to check.